L(s) = 1 | − 2.75i·3-s + (−2.54 + 4.30i)5-s − 3.84·7-s + 1.41·9-s + 6.19·11-s + 16.1·13-s + (11.8 + 7.01i)15-s + 5.20i·17-s − 36.2·19-s + 10.6i·21-s − 22.0·23-s + (−12.0 − 21.9i)25-s − 28.6i·27-s + 20.0i·29-s + 26.4i·31-s + ⋯ |
L(s) = 1 | − 0.918i·3-s + (−0.509 + 0.860i)5-s − 0.549·7-s + 0.157·9-s + 0.562·11-s + 1.23·13-s + (0.789 + 0.467i)15-s + 0.306i·17-s − 1.90·19-s + 0.504i·21-s − 0.958·23-s + (−0.480 − 0.876i)25-s − 1.06i·27-s + 0.690i·29-s + 0.852i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7757654036\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7757654036\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.54 - 4.30i)T \) |
good | 3 | \( 1 + 2.75iT - 9T^{2} \) |
| 7 | \( 1 + 3.84T + 49T^{2} \) |
| 11 | \( 1 - 6.19T + 121T^{2} \) |
| 13 | \( 1 - 16.1T + 169T^{2} \) |
| 17 | \( 1 - 5.20iT - 289T^{2} \) |
| 19 | \( 1 + 36.2T + 361T^{2} \) |
| 23 | \( 1 + 22.0T + 529T^{2} \) |
| 29 | \( 1 - 20.0iT - 841T^{2} \) |
| 31 | \( 1 - 26.4iT - 961T^{2} \) |
| 37 | \( 1 - 69.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.6T + 1.68e3T^{2} \) |
| 43 | \( 1 + 25.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 66.1T + 2.20e3T^{2} \) |
| 53 | \( 1 + 39.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 27.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 54.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 107. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 70.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 37.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 97.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 133.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 6.40iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.845049922278007477467888208603, −8.594143706841403269010175223457, −8.121819885615748642959299326181, −7.06602694857099325205876034740, −6.46690783219183242386959637612, −6.07209742946390982102297042553, −4.28624804976275783170444423883, −3.64044261992270587733473593515, −2.43245263854648656088701907647, −1.31042164167111616599979826516,
0.23413593052594394736996440815, 1.71540058195261368200514881303, 3.38828809106639770869227924763, 4.19586239788753481802817143690, 4.59019906098606552490850089919, 5.96851733633958224039313668464, 6.52829171825969817283999394447, 7.937689819205645301645106377438, 8.458005779465093105395893527943, 9.483973056942914486079639338410